If you look at it abstractly, the ratio between the edge and the area remains the same when we measure it because simply measuring the note does not change the note itself, but on the measurement side, the unit we use to measure the note has an inconsistent ratio between area and edge. Change the unit of measure and you change the ratio in the measurements. But this disturbs me as the area in comparison to the edge doesn't actually change, we just changed unit size.

Thus I am trying to figure out a way to measure area in such a way that the ratio between the area unit and the linear unit remains the same regardless of unit size.

Basically,
If a Post-it note edge (X) is one edge times the unit (U) size, and the Post-it note area (Y) is one area times the unit size, then I want the ratio between X*U and Y*U to remain the same regardless of what U is.

It may require a vastly different way of thinking about area and volume and linear measures, but I'm working on it.

Take a square, an edge of 2 equals an area of 4 thus the ratio of area to edge is 2:1, while an edge of 3 equals an area of 9 and thus the ratio of area to edge is 3:1.

What bothers me is the ratio changed due entirely to unit size. The square didn't change size or shape, so why should the ratio between area and edge change? The area didn't change, the edge didn't change, thus from an abstract view, the ratio between area to edge should not change just because our unit size does.

If changing the unit size changes the ratio, then we can make the ratio whatever we want by changing the unit with which we measure, which then means the ratio is then illusory and useless in that format. It changes despite the fact that the thing measured doesn't change.

It is also means that area vs linear comparisons are non-universal. You can have two people measure the same object and derive the ratio and each will give different results despite measuring the exact same thing.

This non-universalism is my problem. I want to fix it so that the ratio between area and edge remains the same regardless of unit size.

This is actually the very reason that the ratio changes. Because the area goes up as the square of the side.

When you say that a square of side 1 has an area of side 1, this is also not a complete statement. The full statement is: A square of side 1 unit has an area of 1 squared unit.

So if the side is made up of one big block, the area is one big block. But if you chop each side in half, so that each side is 2 half-units, there are now FOUR square half-units. Because area goes up as the square of the side.

Just draw a square and then draw the horizontal and vertical half lines. There are FOUR smaller squares because:

* If you only bisected the x-axis, in other words you turned a square into two rectangles, then there would be TWO tall bricks side by side.

If you only bisected the y-axis, that is you turned a square into two rectangular bricks one on top of the other, there would be TWO bricks in a stack.

If you applied BOTH bisecting operations, first you'd cut the square into two rectangular bricks; and THEN you would cut THOSE BRICKS into two MORE smaller square bricks.

Of which there would be four.

Well this is how I think about it. Perhaps a physicist or a philosopher has a deeper answer.

If you are not convinced by what I wrote, my other piece of advice would be to not worry about it too much. It's just how length and area work.

The ways it works now makes sense, but it has a disadvantage you could say.

I want to make an alternative method that solves that disadvantage, even at the cost of having a different disadvantage.

Just like you might pick different cars based on prioritizing speed or cargo space, so too I want an alternative that has a different goal, namely, maintaining the ratio between area and edge.

You would be going contrary to nature. The "ratio effect" if we call it that is bound up with physics and even biology.

In three dimensions, as the size of an object gets larger, the ratio of its volume to its surface area increases. That's why there could not be giant insects as in horror films. Their exoskeletons would not support their weight. The strength of a skeleton is proportional to the cross-sectional area of the bones or cartilage. But weight is a function of volume.

That's why larger creatures have internal skeletons.

So you see that this ratio thing, which seems like a quirk of abstract math, is actually part of the very fabric of life. Small creatures are more sensitive to cold because their ability to retain heat is a function of their volume; but they lose that heat through their skin, which is measured by their surface area.

So when you are thinking about how to approach your problem, you need to account for giant insects and shivering mice.

Take Post-it notes, we can measure in millimeters and get a different number and thus different ratio than if we measure the same Post-it note in centimeters. Same object, same size, different measurement units, different ratios.

There is a distinction between what math can do and the real world, but that distinction has no bearing on this topic, because the topic is about our measurement and observations.

Take Post-it notes, we can measure in millimeters and get a different number and thus different ratio than if we measure the same Post-it note in centimeters. Same object, same size, different measurement units, different ratios.

There is a distinction between what math can do and the real world, but that distinction has no bearing on this topic, because the topic is about our measurement and observations.

You can see that one square equals four quarter squares, right? I don't mean to discourage your explorations. Maybe you'll find something interesting.

If you look at it abstractly, the ratio between the edge and the area remains the same when we measure it because simply measuring the note does not change the note itself, but on the measurement side, the unit we use to measure the note has an inconsistent ratio between area and edge. Change the unit of measure and you change the ratio in the measurements. But this disturbs me as the area in comparison to the edge doesn't actually change, we just changed unit size.

Thus I am trying to figure out a way to measure area in such a way that the ratio between the area unit and the linear unit remains the same regardless of unit size.

Basically,
If a Post-it note edge (X) is one edge times the unit (U) size, and the Post-it note area (Y) is one area times the unit size, then I want the ratio between X*U and Y*U to remain the same regardless of what U is.

It may require a vastly different way of thinking about area and volume and linear measures, but I'm working on it.

Does anyone have any ideas on how this might be achieved?

(1, 1)
(2, 4)
(3, 9)
.
.
.
The ratio between the original side length and the number of subdivisions is preserved.
1:2 = 2:4, 1:3 = 3:9 and so on.

Ok, from the begining I was thinking this would require an alltogether different approach.

One idea I have at the moment, is to literally make a ratio between the X and Y axi, but this only works for rectangular areas, and would likely be simple to adapt to quadrulaterals by somehow modifying the ratio by the angle of the sides.

But I'm not sure how to apply this to other shapes, such as circles/ovals, polygons of various sizes, or even irregular polygons, etc. Though for the latter, defining a set length of comparison to make the ratio with would also be important, does one use the longest side, or the diameter of a circumscribing circle? etc (I'm inclined towards the circumscribing circle as that is more universal, and if a way to make this work is found for circles, then taking the circumscribing circle and cutting away the excess may be the way to go.).